Automatic constraint satisfaction problem

TL;DR

This paper introduces automata-based constraint satisfaction problems ($AutCSP$), exploring their complexity and extending classical dichotomy results to automata-defined constraints, enabling concise representations of complex problems.

Contribution

It defines $AutCSP$, extends Schaefer's Dichotomy Theorem to automata-based CSPs, and provides polynomial algorithms for tractability over finite domains.

Findings

Checking polymorphisms is polynomial-time for automata-based constraints.

Schaefer's Dichotomy extends to $AutCSP$ over Boolean domains.

Algorithms decide tractability for certain classes of $AutCSP$ instances.

Abstract

We study constraint satisfaction problems (CSPs) where the constraint languages are defined by finite automata, giving rise to automata-based CSPs. The key notion is the concept of Automatic Constraint Satisfaction Problem ($AutCSP$), where constraint languages and instances are specified by finite automata. The $AutCSP$ captures infinite yet finitely describable sets of relations, enabling concise representations of complex constraints. Studying the complexity of the $AutCSP$s illustrates the interplay between classical CSPs, automata, and logic, sharpening the boundary between tractable and intractable constraints. We show that checking whether an operation is a polymorphism of such a language can be done in polynomial time. Building on this, we establish several complexity classification results for the $AutCSP$. In particular, we prove that Schaefer's Dichotomy Theorem extends to the $AutCSP$ over the Boolean domain, and we provide algorithms that decide tractability of some classes of $AutCSP$s over arbitrary finite domains via automatic polymorphisms. An important part of our work is that our polynomial-time algorithms run on $AutCSP$ instances that can be exponentially more succinct than their standard CSP counterparts.